scholarly journals Solvability of nonlinear fractional integro-differential equation with nonlocal condition

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Sakhri Aicha ◽  
Ahcene Merad
Keyword(s):  
Differential Equation ◽  
Functional Analysis ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
A Priori ◽  
Nonlocal Conditions ◽  
Analysis Method ◽  
Content Type ◽  
Fractional Order Differential Equations ◽  
Estimate Method

PurposeThis study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results.Design/methodology/approachThe functional analysis method is the a priori estimate method or energy inequality method.FindingsThe results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions.Research limitations/implicationsThe authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order.Originality/valueThe authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.

scholarly journals Existence and Uniqueness of Abstract Stochastic Fractional-Order Differential Equation

2019 ◽  
Vol 16 ◽  
pp. 8280-8287
Author(s):  
Mahmoud Mohammed Mostafa El-Borai ◽  
A. Tarek S.A.
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Stochastic Analysis ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
Analysis Techniques ◽  
Fractional Order Differential Equations

In this paper, the existence and uniqueness about the solution for a class of abstract stochastic fractional-order differential equations                                           where  in and  are given functions, are investigated, where the fractional derivative is described in Caputo sense. The fractional calculus, stochastic analysis techniques and the standard $Picard's$ iteration method are used to obtain the required.

scholarly journals On a class of nonclassical hyperbolic equations with nonlocal conditions

2002 ◽  
Vol 15 (2) ◽  
pp. 125-140 ◽  
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Continuous Dependence ◽  
Hyperbolic Equations ◽  
Initial Conditions ◽  
A Priori ◽  
Nonlocal Boundary ◽  
Nonlocal Conditions ◽  
A Priori Estimate ◽  
Analysis Method ◽  
Priori Estimate ◽  
Abstract Formulation

This paper proves the existence, uniqueness and continuous dependence of a solution of a class of nonclassical hyperbolic equations with nonlocal boundary and initial conditions. Results are obtained by using a functional analysis method based on an a priori estimate and on the density of the range of the linear operator corresponding to the abstract formulation of the considered problem.

scholarly journals Existence of the Mild Solutions for Impulsive Fractional Equations with Infinite Delay

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Jaydev Dabas ◽  
Archana Chauhan ◽  
Mukesh Kumar
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fractional Order ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Fractional Equations ◽  
Evolution Differential Equation

This paper is concerned with the existence and uniqueness of a mild solution of a semilinear fractional-order functional evolution differential equation with the infinite delay and impulsive effects. The existence and uniqueness of a mild solution is established using a solution operator and the classical fixed-point theorems.

scholarly journals Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Azizollah Babakhani ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Uniqueness Of Solution ◽  
Fractional Order Differential Equations ◽  
Banach Contraction ◽  
Uniqueness Of The Solution

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations(Dα-ρtDβ)x(t)=f(t,x(t),Dγx(t)),t∈(0,1)with boundary conditionsx(0)=x0,  x(1)=x1or satisfying the initial conditionsx(0)=0,  x′(0)=1, whereDαdenotes Caputo fractional derivative,ρis constant,1<α<2,and0<β+γ≤α. Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions onf.

scholarly journals Controllability Analysis for Impulsive Integro-differential Equation via AB Fractional Derivative

2021 ◽  
Author(s):  
KALIMUTHU KALIRAJ ◽  
E. Thilakraj ◽  
Ravichandran C ◽  
Kottakkaran Nisar
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Point Theorem ◽  
Nonlocal Conditions ◽  
Banach Fixed Point Theorem ◽  
Integro Differential Equation

In this work, we analyse the controllability for certain classes of impulsive integro - differential equations(IIDE) of fractional order via Atangana Baleanu derivative involving finite delay with initial and nonlocal conditions using Banach fixed point theorem.

scholarly journals A Refinement of Quasilinearization Method for Caputo's Sense Fractional-Order Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Coşkun Yakar ◽  
Ali Yakar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Unique Solution ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Quasilinearization Method ◽  
Fractional Order Differential Equation ◽  
Fractional Order Differential Equations ◽  
Quasilinearization Technique

The method of the quasilinearization technique in Caputo's sense fractional-order differential equation is applied to obtain lower and upper sequences in terms of the solutions of linear Caputo's sense fractional-order differential equations. It is also shown that these sequences converge to the unique solution of the nonlinear Caputo's sense fractional-order differential equation uniformly and semiquadratically with less restrictive assumptions.

A Strong Solution of an Evolution Problem with Integral Conditions

2002 ◽  
Vol 9 (1) ◽  
pp. 149-159
Author(s):  
S. Mesloub ◽  
A. Bouziani ◽  
N. Kechkar
Keyword(s):  
Functional Analysis ◽  
Parabolic Equation ◽  
Strong Solution ◽  
Existence And Uniqueness ◽  
Energy Inequality ◽  
Integral Boundary Conditions ◽  
Analysis Method ◽  
Integral Conditions ◽  
Integral Boundary ◽  
Singular Parabolic Equation

Abstract The paper is devoted to proving the existence and uniqueness of a strong solution of a mixed problem with integral boundary conditions for a certain singular parabolic equation. A functional analysis method is used. The proof is based on an energy inequality and on the density of the range of the operator generated by the studied problem.

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